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Full text of "Treatise On Analysis Vol-Ii"

30 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (c) Deduce from (a) and (b) that A ^ I/log 4, ^ 5 I/log 2 N (use the fact that £ 1/(N + r) tends to log 2 as N -»• + <x>). r=l (d) Take an = Iog2(2n - 1) - [Iog2(2w - 1)]. Show that we have A = I/log 4 and p, = i/iog 2 for this sequence. (Observe that, if N = 2P, the set {aif..., <ZN} is the same as the set {CN ,..., c2N -1} where cn = Iog2 n — [Iog2 «].) (Cf. Section 13.21, Problem 15.) 7. Let E be a metrizable space. Show that a nonempty closed subset S of E is the support of a lower semicontinuous (resp. continuous) real-valued function ^0 on E if and only if S is the closure of its interior. 8. Let E be a metrizable space and / a mapping of E into ft. The lower semicontinuous regularization of/is defined to be the upper envelope of the continuous functions g on E such that g<*f. (a) For each x e E we denote by lim inf f(y) the greatest lower bound of the numbers y-*x lim inf/(>>„), for all sequences (yn) tending to x Show that the function ;eh->-lim inf/0>) n-»oo y-*x is the lower semicontinuous regularization of/. (b) Suppose that E is an open set in R. For each x e E, let lim inf f(y) denote the y-»x, y£x greatest upper bound of the numbers lim inf/OB) for all sequences (yn) such that n-*oo yn -»• x and yn ^> x for all n. Show that the set of all x e E such that lim inf f(y) > lim inff(y) y-*x, y&x y-+x is at most denumerable. (For each pair (p, q) of rational numbers with p>g, show that the set of points x e E such that lim inf f(y) >p>q>\im Mf(y) y-+xt y&x y-*x is at most denumerable, by following the method of Section 3.9, Problem 3.) 9. A Dirichlet series is a series whose general term is of the form ane~*n5, where (An) is an increasing sequence of real numbers, tending to + oo, and (an) is any sequence of complex numbers, and s is complex. (a) Show that if the series is convergent for 5- = s0, then it is uniformly convergent in the angular sector consisting of the points s = s0 + pew with p ^> 0 and — a £ 6 <i a, where a is such that 0 < a < 7r/2. (Reduce to the case s0 — 0; show first that if 9ts = or, then \e-**^e~**\ g |jio-H*""-«-**) Jb e~~xs dx. For each a integer m and each n^m, put S™, „ — am + am+i H-------h an and use the identity (AbeFs partial summation formula)reater than or equal